Question

All the letters of the word 'EAMCET' are arranged in different possible ways. The number of such arrangements in which two vowels are adjacent to each other, is
(a) 360
(b) 144
(c) 72
(d) 54

Answer 1

Total number of words using letters of EAMCET is
$\begin{array}{rcl}\frac{6!}{2}& =& \frac{6\times 5\times 4\times 3\times 1}{2}\left(\mathrm{Since}\mathrm{E}\mathrm{is}\mathrm{respected}\mathrm{twice}\right)\\ & =& 360\end{array}$
Total number of words with no two vowels together is :-
Since vowels are E, A, E = 3 and consonant is M₁ C₁ T = 3.
i.e number of ways to arrange consonant is 3!
– C – C – C –
i.e vowel can take any of 4 places
i.e number of vowels arrangement = 3! × ⁴P₃ × $\frac{1}{2}$ (∵ E is repeated twice)
$\mathrm{i}.\mathrm{e}3!\times \frac{4!}{1!}\times \frac{1}{2}=72$
∴ Number of words arrangement so that two vowels are adjacent $=\frac{360-72}{2}=144$